Dyson quit his role at the Design Museum in a letter to his fellow trustees, in which he said that the museum in London's Butler's Wharf has become a "style showcase" and abandoned the ideals on which it was founded back in 1989.
Dyson hit out at the organisation, accusing its exhibitions of focusing on style over substance.
Dyson said the final straw came in a string of developments that, in his view, have led to the demise of the museum's strong reputation with the closure of The Conran Foundation Collection, which was replaced by an exhibition on Constance Spry, a 50s flower arranger.
Now Conran has said he might go as well, after commenting he does not know where the museum is going.
"If I am to remain involved, it's got to change. It's not doing the job it was set up to do. It was set up to deal with industrial design, not flower arranging. I may resign if there is not movement, " Conran said.
His criticism echoed that of Dyson, who Conran said he was sad to see go from the museum. He said the situation had become a terrible one.
Dyson has been the chairman of the Design Museum since 1999, when he took over from Conran.
Alice Rawsthorn joined the museum as its new director in 2001, and since then has assumed complete curatorial control. She is behind the move towards having exhibitions such as those of Constance Spry, which are as far removed from industrial design as it is possible to be.
However, Dyson stopped short of criticising Rawsthorn in his stinging letter.
Rawsthorn made no response to the criticisms of the museum raised by Dyson in his letter, and released a diplomatic statement wishing him well.
"We are very grateful to him for leading the museum during this time and thank him for his support and generosity over many years. We wish him every success in his new project."
The appointment of a new chairman will begin later this month.
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